Projected logical ensembles in surface codes via the random-matrix theory of quantum dots

• URL: http://arxiv.org/abs/2606.17140v1
• Date: Mon, 15 Jun 2026 18:00:01 GMT
• Title: Projected logical ensembles in surface codes via the random-matrix theory of quantum dots
• Abstract summary: Measurements underpin active quantum error correction (QEC) and have been recognized as a source of novel measurement-induced many-body phenomena.<n>We study the statistical properties of post-measurement logical states arising in QEC on topological codes subject to deterministic unitary gates.<n>Our results establish a fundamental connection between emergent concepts in mesoscopic physics, quantum many-body systems, and QEC.
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• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Measurements underpin active quantum error correction (QEC) and have been recognized as a source of novel measurement-induced many-body phenomena. Here, we study the statistical properties of post-measurement logical states arising in QEC on topological codes subject to deterministic transversal unitary gates. Upon syndrome extraction followed by maximum-likelihood decoding, a Born-weighted ensemble arises which we dub the "projected logical ensemble" (PLE). Focusing on surface codes subject to uniform single-qubit Pauli-$X$ rotations, we characterize the measurement-induced randomness of the PLE. To this end, we show that for a code with a single logical qubit, the PLE is isomorphic to an ensemble of scattering matrices describing mesoscopic quantum dots obtained from a 2D Majorana network model with suitable boundary conditions. We uncover regimes where these quantum dots are chaotic such that their scattering matrices are well-described by random matrix theory. In these regimes, the PLE approaches a universal ensemble that is maximally random up to symmetry and decoder-induced constraints. The symmetry constraints, set by stabilizer and logical operator weights, realize Altland-Zirnbauer classes D or DIII, which we both illustrate. Our results establish a fundamental connection between emergent universality concepts in mesoscopic physics, quantum many-body systems, and QEC.

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